This section provides the notational conventions used throughout this thesis.
A digital image on a 2D finite arithmetic lattice is a function that maps the supporting lattice onto a finite set of signal values (e.g., grey levels, colours, or multi-band signatures). For simplicity, the set is assumed to be integer-valued in the range between 0 and .
A pixel in an image is denoted by where is the positioning index in the lattice . The set of all pixels in an image with exception of a pixel is denoted by . Similarly, all sites on lattice except the th site is denoted by , so that .
The notation defines a neighbourhood of a site , which usually consists of a set of sites with a specific spatial configuration. All sites in a neighbourhood are dependent of each other.
Because the thesis deals mainly with probability texture models, in particular Markov-Gibbs random fields, related terminology and notations are presented below.
A random variable has a nondeterministic value with a given probability distribution. A sequence or array of statistically interrelated random variables form a discrete stochastic process. Let a random variable be associated with each site . Then these variables form a random field in a configuration space if a probability measure exists. A random field is a two-dimensional stochastic process. The configuration space has a combinatorial number of different images, .
Under a random field model, an image is considered as an instance or a sample of the configuration space . The image probability , or simply, , is known as the joint probability of the image signals. Another commonly-used probability measure is conditional probability of a pixel given all other pixels , denoted by . The conditional probability is a local probability of image signals, while the joint probability is a global one.
For convenience's sake, the notational conventions and symbols used in this thesis are summarised in Table 4.1.
|A 2D finite arithmetic lattice|
|The set of integral signal values|
|A digital image defined on|
|All pixels in the image except the pixel|
|A random field|
|The configuration space of a random field|
|Neighbourhood of the site|
|The lattice with exception of the th site|
|All sites of with exception of a site|
|Joint probability of a sample image|
|Conditional probability of image signal at pixel given|